Problem 15 - Elementary and Intermediate Algebra

Question

Complete each factorization. \(15 c^{3} d^{4}-25 c^{2} d^{4}+5 c^{3} d^{6}=\quad\left(3 c-5+c d^{2}\right)\)

Step-by-Step Solution

Answer

The factorization of the expression is \( 5c^2d^4(3c - 5 + cd^2) \).

Step 1: Identify Common Factors

Examine each term in the expression \( 15c^3d^4 - 25c^2d^4 + 5c^3d^6 \). Identify the greatest common factor among the terms. Here, the common factors in all terms include the coefficients and variables with the lowest power: \( 5c^2d^4 \).

Step 2: Extract the Common Factor

Factor out \( 5c^2d^4 \) from each term. This simplifies as follows:\[ 15c^3d^4 - 25c^2d^4 + 5c^3d^6 = 5c^2d^4(3c - 5 + cd^2) \].

Step 3: Verify the Factorization

Distribute \( 5c^2d^4 \) across the binomial \( (3c - 5 + cd^2) \) to check if the original expression can be re-obtained:- \( 5c^2d^4 \times 3c = 15c^3d^4 \),- \( 5c^2d^4 \times (-5) = -25c^2d^4 \),- \( 5c^2d^4 \times cd^2 = 5c^3d^6 \).The result successfully returns us to the original expression, confirming the factorization.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a key concept in simplifying expressions. In algebra, the GCF of a set of terms is the largest factor that all the terms share. Identifying the GCF is the first step in the process of factorization.
To find the GCF, we:

• Look at the coefficients of the terms. For example, between 15, 25, and 5, the GCF is 5.
• Examine each variable separately and choose the smallest power for each variable present in all terms. For instance, in the expression \(15c^3d^4 - 25c^2d^4 + 5c^3d^6\), the common variable terms are \(c\) and \(d\). The smallest power of \(c\) is \(c^2\) and similarly, for \(d\), it is \(d^4\).

Thus, the GCF for this expression is \(5c^2d^4\). Identifying the GCF helps in breaking down the expression into simpler parts, making it easier to handle and solve mathematical problems.

Distributed Property

The Distributed Property, also known as distribution, is a crucial concept in algebra that allows for the multiplication of a factor over a sum or difference inside parentheses. It states that:
\[ a(b+c) = ab + ac \]
This property comes into play after identifying and factoring out the greatest common factor.
In our example, after finding the GCF \(5c^2d^4\), the expression \(15c^3d^4 - 25c^2d^4 + 5c^3d^6\) becomes:

• \(5c^2d^4(3c - 5 + cd^2)\)

To verify the factorization, we use the Distributed Property:

• Multiply \(5c^2d^4\) by each term inside the parentheses: \(3c\), \(-5\), and \(cd^2\).

This results in the original expression, confirming the factorization is correct. Understanding distribution ensures that factorization unravels the expression back to its original state, verifying its accuracy.

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operation symbols. They form the foundation of algebra, enabling us to describe relationships and patterns.
An algebraic expression can vary in complexity from simple to multi-termed. In our task, the expression \(15c^3d^4 - 25c^2d^4 + 5c^3d^6\) involves three terms.
Here’s what an algebraic expression may include:

• **Constants**: Numbers without variables, like \(-25\) in \(-25c^2d^4\).
• **Variables**: Symbols representing numbers, like \(c\) and \(d\).
• **Coefficients**: Numbers multiplied by variables, such as 15 in \(15c^3d^4\).

Mastering algebraic expressions is fundamental since they are integral to equations and functions found in a plethora of mathematical applications. Through factorization, we simplify expressions, making them easier to work with and more insightful regarding their mathematical properties.